On the Zeros of Linear Factors of Cyclotomic Polynomials Over Galois Fields
John Joseph O. Ondiany *
Department of Mathematics and Actuarial Sciences, Kisii University , P. O. Box 1957-40200, Kisii, Kenya.
Lao Hussein Mude
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
*Author to whom correspondence should be addressed.
Abstract
The study of cyclic codes via cyclotomic polynomial over Galois field has been active area of research due to their direct application in generating cyclic codes and error-correcting codes. Let q be a prime number and Fq be a given finite field with q elements. This research investigates the cyclotomic polynomial yn −1 specifically focusing on cases where yn −1 completely decomposes into linear factors over Fq for q ≤ 37 and n ≥ 2. The relationships between the field, the sum, and the product of the zeros of these linear factors are explored. The results shows that for each tested pair (q,n) where n | (q−1), the sum of the roots is always ≡ 0 (mod q), the product of the roots is ≡ −1 (mod q) and the inverse of the ratio of the product of the roots to n is ≡ q−n (mod q). The predictable modular relationships among the zeros, can be applied to the efficient design of generator polynomials with desired properties.
Keywords: Primitive root of unity, zeros, cyclotomic polynomials